Reference. Decalf: A Directed, Effectful Cost-Aware Logical Framework

Harrison Grodin, Yue Niu, Jon Sterling, Robert Harper · POPL · 2024 · cost-analysis effects · PDF · DOI · arXiv · pldb
We present decalf , a d irected, e ffectful c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs with effects. Like calf , the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike calf , the present language takes account of effects , such as probabilistic choice and mutable state. This extension requires a reformulation of calf ’s approach to cost accounting: rather than rely on a “separable” notion of cost, here a cost bound is simply another program . To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a cost recurrence and readily extends to higher-order, effectful programs. The development proceeds by first introducing the decalf type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify decalf via a model in the topos of augmented simplicial sets.

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@article{grodin-2024-decalf, title={Decalf: A Directed, Effectful Cost-Aware Logical Framework}, volume={8}, ISSN={2475-1421}, url={http://dx.doi.org/10.1145/3632852}, DOI={10.1145/3632852}, number={POPL}, journal={Proceedings of the ACM on Programming Languages}, publisher={Association for Computing Machinery (ACM)}, author={Grodin, Harrison and Niu, Yue and Sterling, Jonathan and Harper, Robert}, year={2024}, month=Jan, pages={273–301} }
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grodin-2024-decalf:
  type: article
  title: 'Decalf: A Directed, Effectful Cost-Aware Logical Framework'
  author:
  - Grodin, Harrison
  - Niu, Yue
  - Sterling, Jonathan
  - Harper, Robert
  date: 2024-01
  page-range: 273-301
  url: http://dx.doi.org/10.1145/3632852
  serial-number:
    doi: 10.1145/3632852
    issn: 2475-1421
  parent:
    type: periodical
    title: Proceedings of the ACM on Programming Languages
    publisher: Association for Computing Machinery (ACM)
    issue: POPL
    volume: 8
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Cited by (4)

Handling Higher-Order Effectful Operations with Judgemental Monadic Laws yang-2026-handlingPOPL · 2026
This paper studies the design of programming languages with handlers of higher-order effectful operations - effectful operations that may take in computations as arguments or return computations as output. We present and analyse a core calculus with higher-kinded impredicative polymorphism, handlers of higher-order effectful operations, and optionally general recursion. The distinctive design choice of this calculus is that handlers are carried by lawless raw monads, while the computation judgements still satisfy the monadic laws judgementally. We present the calculus with a logical framework and give denotational models of the calculus using realizability semantics. We prove closed-term canonicity and parametricity for the recursion-free fragment of the language using synthetic Tait computability and a novel form of the ⊤⊤-lifting technique.
PDF · DOI · arXiv · pldb

Mechanizing Synthetic Tait Computability in Istari li_etal_2025CPP · 2026
Categorical gluing is a powerful technique for proving meta-theorems of type theories such as canonicity and normalization. Synthetic Tait Computability (STC) provides an abstract treatment of the complex gluing models by internalizing the gluing category into a modal dependent type theory with a phase distinction. This work presents a mechanization of STC in the Istari proof assistant. Istari is a Martin-Löf-style extensional type theory with equality reflection, which avoids much of the explicit transport reasoning typically found in intensional proof assistants. This work develops a reusable library for synthetic phase distinction, including modalities, extension types, and strict glue types, and applies it to two case studies: (1) a canonicity model for dependent type theory with dependent products and booleans with large elimination, and (2) a Kripke canonicity model for the cost-aware logical framework. Our results demonstrate that the core STC constructions can be formalized essentially verbatim in Istari, preserving the elegance of the on-paper arguments while ensuring machine-checked correctness.
PDF · DOI · arXiv · pldb

Denotational Foundations for Expected Cost Analysis amorim_2025_oopslaOOPSLA · 2025

Reasoning about the cost of executing programs is one of the fundamental questions in computer science. In the context of programming with probabilities, however, the notion of cost stops being deterministic, since it depends on the probabilistic samples made throughout the execution of the program. This interaction is further complicated by the non-trivial interaction between cost, recursion and evaluation strategy.

In this work we introduce cert: a Call-By-Push-Value (CBPV) metalanguage for reasoning about probabilistic cost. We equip cert with an operational cost semantics and define two denotational semantics — a cost semantics and an expected-cost semantics. We prove operational soundness and adequacy for the denotational cost semantics and a cost adequacy theorem for the expected-cost semantics.

We formally relate both denotational semantics by stating and proving a novel effect simulation property for CBPV. We also prove a canonicity property of the expected-cost semantics as the minimal semantics for expected cost and probability by building on recent advances on monadic probabilistic semantics.

Finally, we illustrate the expressivity of cert and the expected-cost semantics by presenting case-studies ranging from randomized algorithms to stochastic processes and show how our semantics capture their intended expected cost.

PDF · DOI · pldb

Cost-sensitive computational adequacy of higher-order recursion in synthetic domain theory niu-2024-costENTCS · 2024
We study a cost-aware programming language for higher-order recursion dubbed 𝐏𝐂𝐅𝖼𝗈𝗌𝗍 in the setting of synthetic domain theory (SDT). Our main contribution relates the denotational cost semantics of 𝐏𝐂𝐅𝖼𝗈𝗌𝗍 to its computational cost semantics, a new kind of dynamic semantics for program execution that serves as a mathematically natural alternative to operational semantics in SDT. In particular we prove an internal, cost-sensitive version of Plotkin’s computational adequacy theorem, giving a precise correspondence between the denotational and computational semantics for complete programs at base type. The constructions and proofs of this paper take place in the internal dependent type theory of an SDT topos extended by a phase distinction in the sense of Sterling and Harper. By controlling the interpretation of cost structure via the phase distinction in the denotational semantics, we show that 𝐏𝐂𝐅𝖼𝗈𝗌𝗍 programs also evince a noninterference property of cost and behavior. We verify the axioms of the type theory by means of a model construction based on relative sheaf models of SDT.
DOI
Cites 43 works (8 here)
With notes (8)

Quotients, inductive types, and quotient inductive types fiore-2022-quotientsLMCS · 2022
This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.
DOI · arXiv

A Cubical Language for Bishop Sets sterling-2022-aLMCS · 2022
We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets à la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.
DOI

A cost-aware logical framework niu-2022-aPOPL · 2022
We present calf, a cost-aware logical framework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions, we argue that the cost structure of programs motivates a phase distinction between intension and extension. Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, calf presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate calf as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the method of recurrence relations and physicist’s method for amortized analysis. We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid’s algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and parallel complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.
PDF · DOI · arXiv · pldb

Logical Relations as Types: Proof-Relevant Parametricity for Program Modules sterling_harper_2021JACM · 2021

The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction, computational effects, and type abstraction. We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis.

Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure. Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof-relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.”

Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types, by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

DOI · arXiv

Modalities in homotopy type theory rijke-2020-modalitiesLMCS · 2020
Univalent homotopy type theory (HoTT) may be seen as a language for the category of ∞-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a “localization” higher inductive type. This produces in particular the (𝑛-connected, 𝑛-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions.
DOI · arXiv

Cubical Syntax for Reflection-Free Extensional Equality sterling-2019-cubicalFSCD · 2019
We contribute XTT, a cubical reconstruction of Observational Type Theory [Altenkirch et al., 2007] which extends Martin-Löf’s intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity proofs principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel extension of the logical families or categorical gluing argument inspired by Coquand and Shulman [Coquand, 2018; Shulman, 2015]: every closed element of boolean type is derivably equal to either true or false.
DOI · arXiv

A type theory for synthetic ∞-categories riehl-2017-aHigher Structures · 2017
We propose foundations for a synthetic theory of (∞,1)-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities - a “local univalence” condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a “dependent Yoneda lemma” that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using “extension types” that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.
DOI · arXiv

Call-By-Push-Value: A Functional/Imperative Synthesis levy-2003-callbypushvalueSemantic Structures in Computation · 2003
DOI
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grodin-2024-decalf reference entries/refs/grodin-2024-decalf/grodin-2024-decalf.hel